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Nonuniversal statistics in MHD turbulence

Final Report Summary - MHDTURB (Nonuniversal statistics in MHD turbulence)

Turbulence is a multi-scale phenomenon ubiquitous in numerous astrophysical phenomena. The magnetohydrodynamic (MHD) description of astrophysical plasmas is often invoked to theoretically model the energy distribution among a huge range of scales (e.g. from $10^{8} \, m$ to $10^{18} \, m$) measured by a power spectrum, which obeys a power-law exponent. According to theory a regime of intermediate scales exists, where the dynamics are universal, i.e they are not affected by the forcing, the large-scales (which are generally flow-dependent) and by the dissipative dynamics which occur at the very small scales.

The precise scaling of the energy spectrum has implications on the prediction of many astrophysical phenomena such as the heating rates of the solar corona (Cranmer and van Ballegooijen, 2003), the acceleration of the solar wind (Verdini and Velli, 2007; McIntosh et al., 2011), the transport of mass and energy into the Earth's magnetosphere (Sundkvist et al., 2005), the dynamics of the interstellar medium (Spangler and Cordes, 1998; Gaensler et al., 2011), etc. These phenomena are intimately related with the Earth's climate and an example is a very recent work which shows that the solar wind triggers lightning on Earth (Scott et al., 2014). However, to date, theory, simulations, and observations have not been able to provide a definitive answer to the power-law scaling of strong MHD turbulent flows (Iroshnikov, 1964; Kraichnan, 1965; Goldreich and Sridhar, 1995; Mueller and Grappin, 2005; Boldyrev, 2005; Mininni and Pouquet, 2007; Podesta et al., 2007; Lee et al., 2010; Beresnyak, 2012). The power-law scaling exponents -2, -5/3, and -3/2 have been observed in various studies. Therefore, universality (i.e. uniqueness) is ambiguous in the regime of intermediate scales of MHD turbulence. In this project, we investigate the origins of the lack of universal statistics in MHD turbulent flows through state-of-the-art numerical simulations using high-end European computing resources via the PRACE (Partnership for advanced computing in Europe).

An important scientific contribution of the MHDTURB project is the clear demonstration of the dependence of the free-evolving MHD turbulent flows on the large scale initial conditions by
a series of papers (Dallas and Alexakis, 2013a,b,c, 2014b). Using high fidelity numerical data we showed that the -2 scaling exponent is not a solution of our system when certain symmetries are broken for highly turbulent flows. However, when strong cross-correlations between the large scales of the velocity field and the small scales of the magnetic field are significant then magnetic discontinuities (see Fig. 1) may be formed corresponding to a -2 scaling of the energy spectrum. Our theory of strong MHD turbulence with a -2 power-law spectrum due to magnetic discontinuities (common in astrophysical observations) is a possible candidate for the interpretation of the observed -2 power-law spectrum in the Jovian magnetosphere (J.Saur et al., 2002). These results point out that memory of initial conditions persists in MHD turbulence and it can influence the turbulent statistics such as the power law scaling of the energy distribution among scales. Our work has recently received a lot of attention from an article of science for the general public which was written by PRACE (2014) entitled ``Powering up the turbulence of magnetic fields''.

Fusion is the process which powers the sun and makes life on Earth possible. Scientific research in fusion plasmas is central in Europe, with large scale facilities, such as JET and ITER, the most powerful
fusion plasma devices in the world. In order to replicate fusion on Earth, these large scale devices have been built. These devices are able to produce temperatures ten times higher than those in the sun to demonstrate that fusion energy has the potential to provide a sustainable solution to European and global energy needs. However, the development of turbulence in these devices can be fatal. So, one of the greatest challenges is how to suppress turbulence in the fusion plasma devices.

In MHD flows with no external driving force, the existence of multiple conservation laws explains the evolution of these flows toward special states of self-organisation (Matthaeus et al., 2012), where turbulence is suppressed. Self-organisation in turbulent flows is the spontaneous creation of large-scale coherent flow out of a sea of smaller scale homogeneous turbulence (Biskamp, 2003). Within the context of the MHDTURB project we demonstrate that even MHD turbulent flows with electromagnetic external drive can be attracted to such states where turbulence is suppressed (Dallas and Alexakis, 2014a). We show that the correlation time of the external forces is controlling the time spent on these states, i.e. for short correlation times the system remains in the turbulent state while as the correlation time is increased the system spends increasingly more time in the self-organised states. As a result, time averaged statistics can significantly be affected by the time spent on these states and therefore the system can exhibit a non-universal behaviour. This result has important practical implications for the suppression of turbulence in fusion plasma devices. Moreover, an increased knowledge of such dynamics in MHD turbulence can shed light on the understanding of some basic phenomena in the solar corona, such as flares and coronal heating, which occur in a region where observations cannot be performed.

Europe's research area and competitiveness stand to gain considerably from projects focused on fundamental research such as the MHDTURB. This project was focused on fundamental issues intimately related mostly to space and energy, which are challenges that Europe is facing with increasing concern in recent years and where significant resources are being invested. An rough figure of these investments are the budgets based on FP7 factsheets for the years 2007 to 2013, which are € 2.7 billion and € 1.43 billion for Euratom and space related research, respectively. These challenges were addressed effectively in the context of European scientific research and technical innovation through the MHDTURB project. Helping young researchers to improve and enhance their talents in fundamental research will provide multiple long-term benefits in these critical areas for the future sustainability and prosperity of Europe.

More details on the above mentioned results can be found at the following web-page which is dedicated to the MHDTURB project funded by the Marie Curie Action Intra-European Fellowships (FP7-PEOPLE-2011-IEF, Project No. 299973).

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Biskamp, D. Magnetohydrodynamic turbulence. Cambridge University Press, 2003.
Boldyrev, S. On the spectrum of magnetohydrodynamic turbulence. The Astrophysical Journal Letters, 626:L37, 2005.
Cranmer, S. R. and van Ballegooijen, A. A. Alfv ́nic turbulence in the extended solar corona: Kinetic effects and proton heating. The Astrophysical Journal, 594:573, 2003.
Dallas, V. and Alexakis, A. Origins of the k −2 spectrum in decaying taylor-green magnetohydrodynamic turbulent flows. Phys. Rev. E, 88:053014, 2013a.
Dallas, V. and Alexakis, A. Structures and dynamics of small scales in decaying magnetohydrodynamic turbulence. Physics of Fluids, 25:105106, 2013b.
Dallas, V. and Alexakis, A. Symmetry breaking of decaying magnetohydrodynamic Taylor-Green flows and consequences for universality. Phys. Rev. E, 88:063017, 2013c.
Dallas, V. and Alexakis, A. Self-organisation and non-linear dynamics in driven magnetohydrodynamic turbulent flows. to appear in Phys. Rev. E (e-print arXiv:1406.3068) 2014a.
Dallas, V. and Alexakis, A. The signature of initial conditions on magnetohydrodynamic turbulence. The Astrophysical Journal Letters, 788:L36, 2014b.
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Matthaeus, W. H., Montgomery, D. C., Wan, M., and Servidio, S. A review of relaxation and structure in some turbulent plasmas: magnetohydrodynamics and related models. Journal of Turbulence, page N37, 2012.
McIntosh, S. W., De Pontieu, B., Carlsson, M., Hansteen, V., Boerner, P., and Goossens, M. Alfvenic waves with sufficient energy to power the quiet solar corona and fast solar wind. Nature, 475:477–480, 2011.
Mininni, P. D. and Pouquet, A. Energy spectra stemming from interactions of Alfven waves and turbulent eddies. Phys. Rev. Lett., 99(25):254502, 2007.
Mueller, W. C. and Grappin, R. Spectral energy dynamics in magnetohydrodynamic turbulence. Phys. Rev. Lett., 95(11):114502, 2005.
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PRACE. Digest, 2014. URL
Scott, C., Harrison, R., Owens, M., Lockwood, M., and Barnard, L. Evidence for solar wind modulation of lightning. Environmental Research Letters, 9:055004, 2014.
Spangler, S. R. and Cordes, J. M. Vlbi measurements of plasma turbulence associated with the cygnus ob1 association. The Astrophysical Journal, 505:766, 1998.
Sundkvist, D., Krasnoselskikh, V., Shukla, P. K., Vaivads, A., Andr ́, M., Buchert, S., and Reme, H. In situ multi-satellite detection of coherent vortices as a manifestation of Alfvenic turbulence. Nature, 436:825–828, 2005.
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